Breach of contract: When a party to the contract fails or refuses to perform his part of the contract or by his act makes it impossible to perform his obligations under the contract, he is said to have committed breach of contract. Breach of contract is a part of contract law. A breach of contract can happen in both a written and an oral contract. Breach of contract gives the aggrieved party many remedies like right to claim damages, file a suit for specific performance, suit for injunction etc. The parties involved in a breach of contract may resolve the issue among themselves, or in a court of law.
For Example : X agreed to sell 100 bags of rice to Y for ₹10,000 to be delivered on 15th April. On 15th April, X refused to deliver the goods. X has committed breach of contract. Y can take action against X
Breach of trust: A breach of trust occurs when someone is entrusted with a certain duty, and then proceeds to breach that duty. Breach of Trust is a part of Criminal Law.   However, every act of breach of trust may not be resulted in a penal offence of criminal breach of trust unless there is evidence of manipulating act of fraudulent misappropriation. An act of breach of trust involves a civil wrong in respect of which the person may seek his remedy for damages in civil courts but any breach of trust with a mens rea (ie. Guilty mind) gives rise to a criminal prosecution as well. 
 
For Example: 1) A failure to act responsibly for someone who has given you something to keep safe, for example money or a company’s secret information:
2) Investors are claiming losses as a result of a breach of trust by the bank.